I'm not a photography expert at all, but I have spent my whole life playing, buying, and most importantly looking at pictures of games of various kinds. I also have painted many miniatures and developed ways to take good pictures of them without spending a lot of money on serious photo equipment over the years. So here are some thoughts from a gamer perspective, which might be closer to where your clients are coming from.

Are these games not being played by humans? Have you been asked to make sure there are no hands in the process of throwing dice or two players pointing at the board and laughing, etc.? Perhaps a wide shot of friends playing the game, or a closer shot of some hands over the board will help make the photos more relatable. It seems to me like the example photos you have in your question are also way too close. I can't see what's going on in the game, what the board looks like, etc. The racing game is definitely clearer than the hex-based... battle game? And that's because it's a wider shot.

The lighting and white point definitely seem off on your photos. Sunlight can be pretty blue (I suppose it's really the sky) and the shadows can be harsh, so even though it's super-bright, by itself it's not always the best light source. One very cheap way to give yourself more lighting options is to buy a decent sized piece of white foam core, and optionally cover one side with the shiny side of aluminum foil. Especially in sunlight having a reflector can help a lot and provide better color than fill flash (which you might also play around with). You can also just buy a bunch of lamps. They don't have to be expensive photographic lights. You can get cheap LED lights with integral clamps and goosenecks, and you can use tabletop lamps - optionally with the shades taken off. You might really think about the white point of typical household lamps and either try to get consistent white points that you can balance out with camera settings or processing, or you might try mixing white points to get a more natural look (like a warm/cool mix). In the end, a warmer look would probably suit photos of games that would often be played in a living room under incandescent lights. Just getting some high wattage LED bulbs can give you a surprising amount of light without dropping lots of cash on photo equipment.

You can also compensate for less light with longer exposure times. Since your subject isn't prone to moving much, you can really keep that shutter open for a good bit, but you'll want a tripod. If you don't want to invest a lot in a tripod it will probably be cheap and lightweight. One trick for getting longer exposures with a cheap tripod without your hand moving the whole deal is to use the timer function. Set it for a three second timer, a long exposure, hit the shutter and then take a few steps back. Don't even walk on the floor around the time the shutter is open because you can move the tripod just by bouncing the floorboards a little. Some extra household lamps, a foam core reflector or three, a cheap tripod, and a little practice and you should be able to get some bright, warm lighting.

Definitely check out what others are doing. I looked at a random assortment of gaming blogs, I see a lot of lower-quality photos (to my eye, at least). But I did come across one where I like the photos because they look inviting, show me what I want to see, and don't have basic problems that distract from the content: http://www.unboxedtheboardgameblog.com I personally don't see advantage in trying to make photos of board games "exciting". I want to see the elements of the game (cards, counters, pieces, boards, etc.) in a clear fashion. For a review of a game, that's basically it. On that front, you might build a three-sided light box with no floor so you can take close-ups of cards, dice, and other elements with a table surface on the bottom and effective lighting. This sequence of Agricola photos is pretty good, IMHO: http://www.unboxedtheboardgameblog.com/2013/10/keeping-it-organised-agricola-e-raptor.html

This is a great example:

For selling a game, it's common to show people enjoying it, with the implied message of "don't you wish these awesome looking people were at your house having a good time playing with you?"